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Circle

Circle. Set of all points equidistant from a given point called the center. The man is the center of the circle created by the shark. Parts of a circle. We name a circle using its center point. ⊙ W is shown W •.

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Circle

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  1. Circle Set of all points equidistant from a given point called the center. The man is the center of the circle created by the shark.

  2. Parts of a circle We name a circle using its center point. ⊙ W is shown W•

  3. Diameter BERadius XFChord (segment) CD orSecant(line)CDTangent (line or segment) YA or YA C. Y . B . .D . X .E A . F

  4. Arcs – minor arc AB MB YXA major arc BMY BAM XYBSector - area formed by two radii and the arc formed by them (green area) M C E Y B X A

  5. ANGLES in a circleCentral angle  ACB (= measure intercepted arc)Inscribed angle  AMB (= ½ measure intercepted arc)Exterior angle  AEB (= ½ difference of intercepted arcs) Interior angle  AWB (= ½ sum of intercepted arcs) m  ACB = m AB m  AEB = ½(m AB – m XY) m  AMB = ½ m AB m  AWB = ½ (m AB + m MY) M C E Y W B X A

  6. SEGMENT Lengths in a circle External angle forms inverse proportions (* NOTE ORDER) PR = PS PR • PQ = PT • PS PT PQ P S T Q R

  7. SEGMENT Lengths in a circlerelated to chords, secants and tangentsInternal angles form proportions (* NOTE ORDER) BE = AE BE • ED = AE • BC EC ED A D E B C

  8. SEGMENT Lengths in a circlerelated to chords, secants and tangentsExternal angles form proportions (* NOTE ORDER) FG = FJ FG • FH = FJ • FJ FJ FH F G H J

  9. Postulates and theorems about circles • The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. • In the same or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

  10. Postulates and theorems about circles • If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. • If one chord is perpendicular bisector of another chord then the first chord is a diameter. • In the same or congruent circles, two chords are congruent if and only if they are equidistant from the center.

  11. Postulates and theorems about circles • If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. • If two inscribed angles of a circle intercept the same arc, then the angles are congruent. • A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

  12. Postulates and theorems about circles • If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. • If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. • If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

  13. Postulates and theorems about circles • If two chords intersect in the interior of a circle then the product of the lengths of the segments of one chord is equal to the p;product of the lengths of the segments of the other chord. • If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.

  14. Postulates and theorems about circles • If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

  15. Postulates and theorems about circles • In a circle of radius r, an arc of degree measure m has arc length equal to (m/360 • 2πr). • In a circle of radius r, where a sector has an arc degree measure of m, the area of the sector is (m/360 •πr2) • The area of a circle is πr2 • The circumference of a circle is 2πr

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